We consider applying the preconditioned conjugate gradient (PCG) method to solve linear systems Ax = b where the matrix A comes from the discretization of second-order elliptic operators. Let (L +)) ?1 (L t +) denote the block Cholesky factorization of A with lower block triangular matrix L and diagonal block matrix. We propose a preconditioner M = (^ L +)) ?1 (^ L t +) with block diagonal matr...

Yang's η pairing operator is generalized to explore off-diagonal long-range order in the Hubbard bilayer with an arbitrary chemical potential. With this operator and a constraint condition on annihilation and creation operators, we construct explicitly eigenstates which possess simultaneously three kinds of off-diagonal long-range order, i.e., the intralayer one and the interlayer one for on-si...

By counting 1’s in the “right half” of 2w consecutive rows, we locate the main diagonal of any doubly infinite permutation matrix with bandwidth w. Then the matrix can be correctly centered and factored into blockdiagonal permutation matrices. Part II of the paper discusses the same questions for the much larger class of band-dominated matrices. The main diagonal is determined by the Fredholm i...

Let A be an n n complex matrix and r be the maximum size of its principal submatrices with no o¤-diagonal zero entries. Suppose A has zero main diagonal and x is a unit n-vector. Then, letting kAk be the Frobenius norm of A; we show that jhAx;xij (1 1=2r 1=2n) kAk : This inequality is tight within an additive term O n 2 : If the matrix A is Hermitian, then jhAx;xij (1 1=r) kAk : This inequality...

The jam phases in a two-dimensional cellular automata model of traffic flow are investigated by computer simulations. Two different types of the jam phases are found. The spatially diagonal long-range correlation obeys the power law at the low-density jam configurations. The diagonal correlation exponentially decays at the high-density jam. The exponent of the short-range correlation in the dia...

subject to the initial conditions A(l, 0) = 1, A(l, 1) ='2, with 4(n, /c) = 0 for & < 0 or k > n. This array has been called a Lucas triangle by Feinberg [1], because rising diagonals sum to give the Lucas numbers 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, ..., in contrast to the rising diagonals in the standard Pascal triangle where rising diagonals sum to give the Fibonacci numbers 1, 1, ...

∀i, yi = [0, ..., 0 } {{ } j−1 , 1, 0, ..., 0 } {{ } c−j ] ⇒ xi W0 = x̄j W0, (1) where y i is the i-th row of the true cluster assignment matrix Y and x̄j is the mean of the data that belongs to class j. Denote X̄c = [x̄1, ..., x̄c]. Note that X̄c = XY Σ, where Σ ∈ Rc×c is a diagonal matrix with the i-th diagonal element as 1/ni, ni is the number of the data that belongs to class i. Then rank(X̄ c W0)...

Simple transformation formulas between fermion matrices and observables, and numerical values of quark matrices, are obtained on a particular weak basis with one quark matrix diagonal and the other with vanishing elements 1-1, 1-3 and 3-1, and with only the element 2-2 complex. When we chooseMu diagonal, thenMd shows intriguing numerical properties which suggest a four parameter description of ...

We study the properties of gluons in QCD in the maximally abelian (MA) gauge. In the MA gauge, the off-diagonal gluon behaves as the massive vector boson with the mass Moff ' 1.2 GeV, and therefore the off-diagonal gluon cannot carry the long-range interaction for r M−1 off ' 0.2 fm. The essence of the infrared abelian dominance in the MA gauge is physically explained with the generation of the...

Alternative definition: Then LG is the matrix with diagonal entries (LG)ii = di, and off-diagonal entries (LG)ij = −wij . Fact 1: The matrix L = LG is symmetric, non-diagonals entries are Lij = −wij , and its diagonal entries are Lii = di, where di = ∑ j:ij∈E wij is the degree of vertex i. It is useful to put these values in a diagonal matrixD = diag(d⃗). If G is unweighted, then L = D−A where A...